Does Goedel's incompleteness theorem hold true for reals?

I have many times wondered whether the incompleteness theorem extends to real numbers, which are usually the stuff used by physics. There is a very nice discussion of this point here. Strongly recommended.

Real numbers and all algebraically closed number fields such as complex numbers and algebraic numbers are complete. All truths are provable. If physics is based on complex numbers or algebraic numbers, Goedel's theorem has no direct implications for physics.This however implies that integers cannot be characterized using the axiomatics of these number fields since if this were the case, Gdel's incompleteness theorem would not hold true for integer arithmetics. One can also say that Goedel numbers for unprovable theorems are not expressible as a natural number but are more general reals or complex numbers.

Since algebraic numbers are complete, a good guess is that algebraic numbers label all true statements about integer arithmetics and also about arithmetics of algebraic integers for extensions of rationals.

In TGD adelic physics definescorrelates for cognition. Adeles for the hierarchy labelled by algebraic extensions (perhaps also extensions involving roots of e since e^p is p-adic number). These are not complete and Goedel's incompleteness theorem applies to them. Only at the never achievable limit of algebraic numbers the system becomes complete. This would strongly suggest a generalization of Turing's view about computation by replacing integer arithmetics with a hierarchy of arithmetics of algebraic integers associated with extensions of rationals. See this article .

For a summary of earlier postings see Latest progress in TGD.

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